With some constants \(a,b,c,d\in {\mathbb {C}}\) , we can write
\(K=a\frac{\psi (\big ||x|+ |y|\big |^2)}{|x|+ |y|}+b\frac{\psi (\big ||x|- |y|\big |^2)}{|x|- |y|}+c\frac{\psi (||x|-|y||^2)}{|x|+ i|y|}+d \frac{\psi (||x|-|y||^2)}{|x|- i|y|}.\) 
We set \(\chi _\pm =\chi _{{\mathbb {R}}_\pm }\) , \(\tau f(x)=f(-x)\)  and
\(\widetilde{k}_1(x,y)=\frac{\psi (|x- y|^2)}{x-y},\quad \widetilde{k}_2^\pm (x,y)=\frac{\psi (|x- y|^2)}{x\pm iy}.\) 
Then \(T_K\)  is written in the form
\(\nonumber T_{K}&=\left\lbrace a\left(\chi _{+}T_{\widetilde{k}_1}\chi _{-}-\chi _{-}T_{\widetilde{k}_1}\chi _{+}\right)+b\left(\chi _{+}T_{\widetilde{k}_1}\chi _{+}-\chi _{-}T_{\widetilde{k}_1}\chi _{-}\right)\right\rbrace (1+\tau )\\&\quad +\left\lbrace c\left(\chi _+T_{\widetilde{k}_2^+}\chi _+- \chi _-T_{\widetilde{k}_2^+}\chi _-\right)+d\left(\chi _+T_{\widetilde{k}_2^-}\chi _+- \chi _-T_{\widetilde{k}_2^-}\chi _-\right)\right\rbrace (1+\tau ).\) 

Indeed, since
\(k_1^\pm (x,y)=\widetilde{k}_1(|x|,\mp |y|)\)  and \(k_1(-x,-y)=-k_1(x,y)\) ,
we have
\(k_1^\pm (x,y)&=\left(\chi _{+}(x)+\chi _{-}(x)\right)k_1^\pm (x,y)\left(\chi _{+}(y)+\chi _{-}(y)\right)\\&=\chi _{+}(x)\widetilde{k}_1 (x,\mp y)\chi _{+}(y)+\chi _{+}(x)\widetilde{k}_1(x,\pm y)\chi _{-}(y)\\&\quad -\chi _{-}(x)\widetilde{k}_1(x,\pm y)\chi _{+}(y)-\chi _{-}(x)\widetilde{k}_1(x,\mp y)\chi _{-}(y).\) 

By the change of variable \(y\mapsto -y\)  in the first and fourth terms for the “\(+\) " case and the second and third terms for the “\(-"\)  case, respectively, we obtain
\(T_{k_1^\pm }f(x)=[(\chi _{+}T_{\widetilde{k}_1}\chi _{\mp }-\chi _{-}T_{\widetilde{k}_1}\chi _{\pm })(1+\tau )]f(x).\) 

A similar calculation also implies
\(k_2^\pm (x,y)&=\chi _+(x)\widetilde{k}_2^\pm (x,y)\chi _+(y)+\chi _+(x)\widetilde{k}_2^\pm (x,-y)\chi _-(y)\\&\quad -\chi _-(x)\widetilde{k}_2^\pm (x,-y)\chi _+(y)-\chi _-(x)\widetilde{k}_2^\pm (x,y)\chi _-(y).\) 

Hence, by the change of variable \(y\mapsto -y\)  in the second and third terms, we have
\(T_{k_2^\pm }f(x)=[(\chi _+T_{\widetilde{k}_2^\pm }\chi _+- \chi _-T_{\widetilde{k}_2^\pm }\chi _-)(1+\tau )]f(x).\) 

These two formulas imply [lemma23proof1]lemma23proof1. Since both the multiplication operator by \(\chi _{\pm }(x)\)  belongs to \(\mathbb {B}(L^p(w_p))\cap \mathbb {B}(L^{1,\infty }(w_1))\)  with operator norms 1 for all \(1\le p<\infty \) , we obtain
\({\Vert T_{K}f\Vert }_{\mathcal {Y}}\lesssim \left({\Vert T_{\widetilde{k}_1}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}+{\Vert T_{\widetilde{k}_2^+}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}+{\Vert T_{\widetilde{k}_2^-}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}\right)\left({\Vert f\Vert }_{\mathcal {X}}+{\Vert \tau f\Vert }_{\mathcal {X}}\right)\) 

if \((\mathcal {X},\mathcal {Y})\in \lbrace (L^p(w_p),L^p(w_p)),(L^1(w_1),L^{1,\infty }(w_1))\rbrace \) . Moreover, since
\(\overline{\widetilde{k}_1(y,x)}=-\widetilde{k}_1(x,y),\quad \overline{\widetilde{k}_2^\pm (y,x)}=\pm i\widetilde{k}_2^\pm (x,y)\) 

we have \((T_{\widetilde{k}_1})^*=-T_{\widetilde{k}_1}\)  and \((T_{\widetilde{k}_2^\pm })^*=\pm iT_{\widetilde{k}_2^\pm }\)  and
\({\Vert T_{K}^*f\Vert }_{\mathcal {Y}}\lesssim \left({\Vert T_{\widetilde{k}_1}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}+{\Vert T_{\widetilde{k}_2^+}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}+{\Vert T_{\widetilde{k}_2^-}\Vert }_{\mathcal {X}\rightarrow \mathcal {Y}}\right)\left({\Vert f\Vert }_{\mathcal {X}}+{\Vert \tau f\Vert }_{\mathcal {X}}\right).\) 
By virtue of Theorem REF , it thus is enough to show that \(T_{\widetilde{k}_1},T_{\widetilde{k}_2^\pm }\)  are Calderón–Zygmund operators, namely \(\widetilde{k}_1,\widetilde{k}_2^\pm \)  are standard kernels and \(T_{\widetilde{k}_1},T_{\widetilde{k}_2^\pm }\in \mathbb {B}(L^2({\mathbb {R}}))\)  (see Appendix ).
Since \(\widetilde{k}_1,\widetilde{k}_2^\pm \)  are supported away from a neighborhood of the diagonal line, they are smooth on \({\mathbb {R}}^2\) . Moreover, since \(1\le |x-y|^2\le 2\)  on \(\mathop {\mathrm {supp}}\nolimits \psi ^{\prime }(|x-y|^2)\) , we have
\(|\partial _x^\alpha \partial _y^\beta \widetilde{k}_1(x,y)|+|\partial _x^\alpha \partial _y^\beta \widetilde{k}_2^\pm (x,y)|&\lesssim {\langle }x-y{\rangle }^{-1-\alpha -\beta },\quad \alpha ,\beta =0,1,2,....\) 

Hence \(\widetilde{k}_1\)  and \(\widetilde{k}_2^\pm \)  are standard kernels. To show \(T_{\widetilde{k}_1}\in \mathbb {B}(L^2({\mathbb {R}}))\) , we observe that, modulo a rapidly decaying error term, \(T_{\widetilde{k}_1}\)  is essentially a truncated Hilbert transform \(\mathcal {H}^{(2)}\)  defined by
\(\mathcal {H}^{({\varepsilon })}f(x)=\int _{|x-y|>{\varepsilon }}\frac{f(y)}{x-y}dy,\quad {\varepsilon }>0.\) 
Indeed, since \(\psi (s)=1\)  for \(s\ge 2\) , we have, for any \(N\ge 0\) ,
\(\widetilde{k}_1(x,y)=(\chi _{\lbrace |x-y|>2\rbrace }+\chi _{\lbrace |x-y|\le 2\rbrace })\widetilde{k}_1(x,y)=\frac{\chi _{\lbrace |x-y|>2\rbrace }}{x-y}+O({\langle }x-y{\rangle }^{-N}).\) 

Since \(\mathcal {H}^{(2)}\in \mathbb {B}(L^2({\mathbb {R}}))\)  (see [1]}) and the error term is also bounded on \(L^2({\mathbb {R}})\)  by Lemma REF , so is \(T_{\widetilde{k}_1}\) . For the operators \(T_{\widetilde{k}_2^\pm }\) , we compute
\(\widetilde{k}_2^\pm (x,y)=\frac{x}{x^2+y^2}\psi (|x- y|^2)\mp i\frac{y}{x^2+y^2}\psi (|x- y|^2)=:\widetilde{k}_{21}(x,y)\mp i \widetilde{k}_{22}(x,y).\) 
Then, \(T_{\widetilde{k}_{21}}\in \mathbb {B}(L^\infty ({\mathbb {R}}))\cap \mathbb {B}(L^1({\mathbb {R}}),L^{1,\infty }({\mathbb {R}}))\)  since
\(\sup _{x\in {\mathbb {R}}}\int |\widetilde{k}_{21}(x,y)|dy\lesssim \sup _{x\in {\mathbb {R}}}\int \frac{|x|}{x^2+y^2+1}dy\lesssim 1,\quad |\widetilde{k}_{21}(x,y)|\lesssim {\langle }x{\rangle }^{-1}\in L^{1,\infty }({\mathbb {R}}).\) 
The Marcinkiewicz interpolation theorem then yields \(T_{\widetilde{k}_{21}}\in \mathbb {B}(L^2({\mathbb {R}}))\) . Since \(T_{\widetilde{k}_{22}}=(T_{\widetilde{k}_{21}})^*\) , \(T_{\widetilde{k}_{22}}\in \mathbb {B}(L^2({\mathbb {R}}))\)  by duality. Hence \(T_{\widetilde{k}_{2}^\pm }\in \mathbb {B}(L^2)\) . Summarizing these arguments, we conclude that \(T_{\widetilde{k}_1}\)  and \(T_{\widetilde{k}_2^\pm }\)  are Calderón–Zygmund operators. This completes the proof.
